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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.airy.bi"></a><a class="link" href="bi.html" title="Airy Bi Function">Airy Bi Function</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.airy.bi.h0"></a>
        <span class="phrase"><a name="math_toolkit.airy.bi.synopsis"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">airy_bi</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>

 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">airy_bi</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.airy.bi.h1"></a>
        <span class="phrase"><a name="math_toolkit.airy.bi.description"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.description">Description</a>
      </h5>
<p>
        The function <a class="link" href="bi.html" title="Airy Bi Function">airy_bi</a> calculates
        the Airy function Bi which is the second solution to the differential equation:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span>

        </p></blockquote></div>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<p>
        The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span>
        changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while
        for positive <span class="emphasis"><em>x</em></span> the value tends to infinity:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.airy.bi.h2"></a>
        <span class="phrase"><a name="math_toolkit.airy.bi.accuracy"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.accuracy">Accuracy</a>
      </h5>
<p>
        This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
        - refer to those functions for detailed accuracy information.
      </p>
<p>
        In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x
        &gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
        0</em></span> as the following error plot illustrate:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/bi__double.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.airy.bi.h3"></a>
        <span class="phrase"><a name="math_toolkit.airy.bi.testing"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.testing">Testing</a>
      </h5>
<p>
        Since this function is implemented in terms of other special functions, there
        are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>.
      </p>
<h5>
<a name="math_toolkit.airy.bi.h4"></a>
        <span class="phrase"><a name="math_toolkit.airy.bi.implementation"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.implementation">Implementation</a>
      </h5>
<p>
        This function is implemented in terms of the Bessel functions using the relations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/airy_bi.svg"></span>

        </p></blockquote></div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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